Cube corner retroreflector with limited range

ABSTRACT

Retroreflective articles having cube corner elements are disclosed, with the dihedral angle errors of the cube corner elements selected to limit the visibility range of the retroreflective article. Also disclosed are methods for making the cube corner elements and the retroreflective articles.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a divisional application claiming priority to U.S.application Ser. No. 11/018,828, filed Dec. 21, 2004, which in turnedclaimed the benefit under 35 U.S.C. § 119(e) of U.S. ProvisionalApplication 60/532,496 filed Dec. 24, 2003, both titled CUBE CORNERRETROREFLECTOR WITH LIMITED RANGE, which applications are herebyincorporated by reference in their entirety.

BACKGROUND

The present invention generally relates to retroreflective articles,and, more specifically, to cube corner retroreflectors havingselectively reduced visibility range.

Retroreflective articles are well-known for applications such as highwaysigns, safety reflectors, and road markers. Generally, cube cornerversions of these articles have a frontal lens of clear, colored oruncolored resin, such as methyl methacrylate, with a smooth frontsurface and a plurality of retroreflective cube corner elements on therear surface. The cube corner elements each have three reflecting faces.

Light from a remote source passes through the smooth front surface,reflects off each of the three faces of a cube corner element, andpasses again through the front surface. In a perfect retroreflector,this light is returned in a direction exactly opposite to the incomingdirection of light. Primarily because of imperfections, eitheraccidental or by design, the reflected light is not returned only in adirection exactly opposite to the incoming direction, but rather isreturned typically into a spreading pattern, centered on the exactreturn direction. This imperfect return reflection is still termed“retroreflection”. The spread retroreflected light enables theretroreflector to be visible from directions slightly away from thelight source.

For example, if headlights from an automobile are the source of light,then the perfect retroreflective pavement marker would reflect lightback only toward the headlights. It is desirable that the reflectedlight from a retroreflective pavement marker be seen by the driver ofthe automobile, whose eyes are generally higher than and somewhat leftor right of the headlights.

Changes to the size or shape of the faces of the cube corner prismelements, or to the angles between the faces (dihedral angles), or tothe flatness of the faces or the flatness of the front surface, can allchange the pattern of retroreflection and thereby determine the regionsaround the light source in which the retroreflector visible. “Study ofLight Deviation Errors in Triple Mirrors and Tetrahedral Prisms”, J.Optical Soc. Amer., vol. 48, no. 7, pp. 496-499 (July 1958) by P. R.Yoder, Jr., describes spot patterns resulting from the angles betweenfaces being not exactly right angles. U.S. Pat. No. 3,833,285, toHeenan, which is incorporated in its entirety herein by reference,teaches that having one dihedral angle of a macro-sized cube cornerelement greater than the other two results in extended observationangularity in macrocubes, and specifically that the retroreflected lightdiverges in an elongated pattern. This elongated pattern has a generallysubstantially vertical axis. U.S. Pat. No. 4,775,219, to Appeldorn etal., teaches redistribution of the reflected light so that more light isdirected to the driver of approaching vehicles or extending the patternof light by modifying the dihedral angles of micro-sized cube corners.

The angle formed between the source, the retroreflector, and theobserver is called the observation angle. Conventional pavement markersand other retroreflective articles are generally designed to be highlyvisible at long distances, corresponding to small observation angles.Because of imperfections, generally accidental, in conventionalretroreflective articles, they are also highly visible at middle andclose distances, corresponding to medium and large observation angles.For each type of retroreflector in each application, the relative valueof long, medium, and close visibility may differ. Some researchers havesuggested that long distance visibility of pavement markers might not beuseful, or even have negative value.

SUMMARY OF THE INVENTION

The present invention provides a retroreflective article having cubecorner prism elements that are constructed to selectively limit therange at which the article is visible. The retroreflection of light fromthe article at small observation angles, for example 0.3 degrees andless, is selectively limited. While the article may still be visible atlong distance, the intensity of the reflected light is limited. This hasapplication, for example, in raised pavement markers, so that the markeris visible to a vehicle driver sufficiently in advance of the marker toenable the driver to react to the marker, but with limited visibility atlarger distances from the marker to avoid excessive road preview andavoid distraction of the driver.

Retroreflective road markers made in accordance with the presentinvention will have limited visibility beyond a certain distance D, andhigh visibility at substantially closer distances. A consequence is thatthe retroreflectance of such articles is not a simply decreasingfunction of observation angle, as is the retroreflectance ofconventional retroreflectors. The retroreflectance of an article made inaccordance with the present invention generally has a peak value at aselected observation angle with decline at observation angles less thanthe selected observation angle to assure more rapid decrease ofvisibility beyond a selected distance than conventional retroreflectorsexhibit.

The variation of retroreflectance with observation angle can be obtainedby deviating the dihedral angles of the cube corner prism elements fromthe orthogonal to obtain cube corner dihedral angle errors. Theintensities of retroreflected light at desired observation angles may beobtained by making the cube corners divergent or convergent sufficientlyto direct the retroreflected light away from the undesired observationangles.

Articles made in accordance with the present invention will havepatterns of retroreflection that are relatively weak in their centers,at the smallest observation angles corresponding to long distances, andstronger away from their centers at the middle observation angles,especially at the relevant Epsilon angles. Such light patternscorrespond to the needs of drivers at moderate distances.

An example to bring about the desired functional coefficients ofluminous intensity is to form the cube corners to produce an efficientlight reflection pattern by molding acrylic resin, for example, toobtain cube corner elements having dihedral angle errors of about 0degrees, −0.13 degrees, and 0 degrees. The third value refers to adihedral edge in an approximately vertical left-right symmetry plane ofthe marker. This would result in an efficient light reflection patternin a raised pavement marker having a 30 degree sloping front.

The cube corner element made from a cube corner tool has three dihedralangle errors corresponding to the three dihedral angle errors in thecube corner tool. The process, such as molding, for forming the cubecorner elements from the cube corner tool generally will cause someshrinkage of the angles in the cube corner elements, which may not beequal for the three dihedral angles. There is a transformation ofdihedral angles, wherein for each of the three dihedral angles theamount of change between the dihedral angle errors of the cube cornerelement in the tool and the dihedral angle errors of the cube cornerelement in the molded part is independent of the dihedral angle errorsin the tool, and substantially constant for repeated applications of theforming process. A retroreflector generally has many cube corners, and,for discussion purposes, it is presumed that the same transformationapplies to all the cube corners in the retroreflector, although this isnot necessarily the case. Applied to the example of the previousparagraph, if the molding process transforms dihedral angle errors byabout −0.04, about −0.04, and about −0.07 degrees, respectively, thenthe desired cube corner article having dihedral angle errors of about 0degrees, −0.13 degrees, and 0 degrees would be made from a tool havingdihedral angle errors of about +0.04 degrees, −0.09 degrees, and +0.07degrees. For some forming processes the transformation amounts arepositive, meaning dihedral angle swelling instead of shrinking, andthere may also be processes where the amounts are zero. The manufacturermust know his process before making tools so that the finalretroreflective part has the desired dihedral angles.

The present invention includes cube corner elements designed to producea pattern of retroreflection to selectively limit the range ofvisibility, the tools to make such cube corner elements, and methods formaking retroreflective articles having such cube corner elements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a sectional elevational schematic view of a raised pavementmarker illustrating one embodiment of the present invention;

FIG. 2 is a geometrical representation of a retroreflective cube cornerelement of the present invention;

FIG. 3 is a geometrical representation of one of the dihedral angles ofa retroreflective cube corner element of the present invention;

FIGS. 4A-D illustrate the dihedral naming convention for four kinds ofcube corner elements;

FIG. 5 is a representation of a spot pattern of reflected light from atypical prior art raised pavement marker;

FIG. 6 is a representation of a spot pattern of reflected light from arotated cube corner of the prior art;

FIG. 7 is a representation of a spot pattern of reflected light from oneembodiment of the present invention;

FIG. 8 is representation of a spot pattern of reflected light fromanother embodiment of the present invention;

FIGS. 9A-D illustrate, respectively, the spot pattern from a cube of atool from Example 1, the spot pattern from a cube from a correspondingmolded part but as-if in a tool, the spot pattern from a cube from thecorresponding molded part in its sloping plastic lens, and the spotpattern from a pairing of mirror image cubes in the molded part;

FIG. 10 illustrates the addition of two spot patterns;

FIG. 11 illustrates the spots corresponding to passenger car observationin the distance range 40 m to 130 m. Imagine that you are behind theleft headlamp viewing the retroreflector. For a reflection to reach thecar driver it must diverge by an amount, and in a direction shown by theswarm of points on the right side of the figure. Similarly for the rightheadlamp and the swarm of points on the left side of the figure;

FIG. 12 illustrates the expected spreading of a point pattern into alight intensity pattern by high quality injection molding;

FIGS. 13A and B illustrate possible ranges for desirable spot patterns;

FIGS. 14A-D are graphs of the performance of the standard marker and amarker made in accordance with Example 1 of the present invention;

FIGS. 15A and B illustrate the spot patterns in accordance withembodiments of the present invention;

FIG. 16 is a graph of the VI2 performance of the standard marker and amarker made in accordance with Example 3 of the present invention;

FIG. 17 illustrates a cube corner design that produces spots withdefinitely unequal light content;

FIG. 18 illustrates the spot patterns from the same molded part asillustrated in FIGS. 8 and 9C, but with the horizontal entrance angle ata non-zero degree value, and

FIG. 19 illustrates how hexagonal cube corners can be assembled in atightly packed array for the marker lens of Example 3.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed to retroreflective cube cornerelements providing selectively limited visibility range, retroreflectivearticles having such cube corner elements, and tools and methods ofmaking such cube corner elements and retroreflective articles. The cubecorner elements may be used in all retroreflective items, such as insheeting for highway signs and reflective vests and in lenses forvehicle reflectors but are particularly well suited for use in raisedpavement markers and other road delineators. The invention is describedin relation to a raised pavement marker retroreflecting the light of theheadlights of a car. Reflectors on vehicles, or other retroreflectiveitems, such as signs, also may be limited in range in the mannerdescribed herein. Specific parameter values may be determined by thoseof ordinary skill in the art without undue experimentation. Theparticular embodiments described are illustrative only, and theinvention is limited only by the claims.

Use of the terms “horizontal” and “vertical” herein in reference toraised pavement markers presumes the marker to be mounted on horizontalpavement. The term “road direction” used herein in reference to a raisedpavement marker means the horizontal direction from which a vehicleextremely far away on a straight road would illuminate the markerproperly mounted on the road. The term “horizontal entrance angle” usedherein is defined as the angle between the direction of light incidenton the pavement marker and the road direction, as projected into ahorizontal plane. The term “observation angle” herein is defined as theangle between a viewer's line of sight to a retroreflector and a linefrom a light source to the retroreflector. This observation anglecorrelates to distance from the retroreflective article. Generally, thefarther the viewer is from the retroreflective article, the smaller theobservation angle will be. At each distance, there are two observationangles for the typical vehicle, corresponding to the two headlights. Theterm “Epsilon angle” used herein is defined as the angle between twolines in the plane passing through the marker lens center and normal tothe road direction. One line is vertical. The other line is theintersection of this plane with the plane containing the observationangle. This Epsilon angle correlates with the driver's position within avehicle. The driver's being slightly right of the vehicle's leftheadlight produces a small positive value for this Epsilon. The driver'sbeing far to the left of the vehicle's right headlight produces a largernegative value for this Epsilon. There are two Epsilons for a typicalvehicle corresponding to the two headlights which make two observationangle planes. The term “the standard marker” used herein refers to amarker having macro cube corner elements with no intentionally-induceddihedral angle errors. Its lens has a forward projected area of 10 cm².Its retroreflective performance is defined by several measuresthroughout this application.

It is conventional to describe the retroreflectance by a coefficient ofluminous intensity, termed “R_(I)” in the International Commission onIllumination (CIE) publication “Retroreflection: Definition andMeasurement” (CIE Pub. 54.2). The R_(I) is the ratio of the luminousintensity leaving the retroreflector, in a certain direction, to theilluminance received at the marker, in a certain direction. Since thisis a retroreflector, the R_(I) is high when the output direction isopposite to the input direction, or nearly so. The relation of inputdirection to output direction may be described with the two angles Alphaand Epsilon as defined in CIE Pub. 54.2. Alpha is observation angle andEpsilon is the tilt of the observation angle. Observation angle measuresthe divergence between the output direction and the input direction. Theinput direction may be described with the two angles Beta and Omega asdefined in CIE Pub. 54.2. The variation of R_(I) with Alpha and Epsilonis called the “pattern of retroreflection.” In order to assign a singleretroreflectance value at a particular observation angle Alpha in thispattern, it is necessary to make some assumptions about the vehiclegeometry. This will produce a “Functional Coefficient of LuminousIntensity at observation angle Alpha” as explained below.

Road markers are almost always used to delineate driving lanes. When aroad marker is at a certain distance from a vehicle, the Alpha anglesfor the left and right headlights can be estimated from the geometry ofthe vehicle and the driving lane. The headlight illuminance received bythe road marker at this distances can be estimated from published dataon headlight beam patterns. Except at very close distances, the left andright headlights provide nearly equal illuminance to markers. Then foreach distance, the luminous intensity of the road marker, in thedirection of the driver, is the product of the functional coefficient ofluminous intensity with the expected illuminance.

Road markers are generally installed in equal interval rows. The mostrelevant visibility is not that of the individual road markers but ofthe section of the row comprising a small number of markers. An estimateof this visibility is nevertheless the luminous intensity of theindividual markers. The estimate will be defined as indices VI1 and VI2as explained below.

Once the input direction and output direction are specified, thecoefficient of luminous intensity R_(I) of a retroreflective marker is adefinite measureable quantity. The input direction is defined as theroad direction. The output direction depends on the two angles Alpha andEpsilon. There is no simple “R_(I) at Alpha” because of the need tospecify Epsilon. To produce a functionally relevant value correspondingto “R_(I) at Alpha”, assumptions are made about the vehicle geometry.The “international” assumption assumes three Epsilons, −45°, 0°, +45°,all applicable to the given Alpha. The “asymmetrical” assumption assumesEpsilon for the vehicle's left headlight to be +20° and Epsilon for thevehicle's right headlight to be −50°, and also producing two Alphavalues, one for each headlight, from the given Alpha. The correspondingtwo definitions FRO1 and FRO2 are given in Table 1. By making furtherassumptions about the vehicle geometry, functionally relevant valuescorresponding to “R_(I) at Distance D” are given in the two definitionsFRD1 and FRD2 in Table 1. The relation of distance to observation angleembodied in these two definitions implies a vehicle larger than anstandard passenger car. The constant 50 in the definition of FRD1 andthe constants 40 and 60 in the definition of FRD2 all have units(metre·degree). It should be emphasized that each functionalretroreflectance value FRO1, FRO2, FRD1, and FRD2 is determineddefinitely from two or three R_(I) measurements on a marker. No furtherdata are required for the determinations.

The ultimate measure of a retroreflector's function is its visibility.For road markers this is appraised as the individual marker'scontribution to the visibility of a row of like markers. Eachretroreflecting marker produces a luminous intensity. The luminousintensity is obtained from the retroreflectance value FRD1 or FRD2 bymultiplication with the headlight's illuminance. The headlightilluminance will be assumed to follow an inverse 2.5 power law fordistance. This decrease with distance is somewhat faster than theinverse square law because with increasing distance not only does theillumination beam spread, but its aim takes it away from the marker. Thefaraway road marker appears nearly as a point, so the luminous intensityshould next be divided by the square of the distance, yielding theilluminance at the eye. Finally, the visual angular separation betweensuccessive faraway road markers is inversely proportional to the squareof the distance. The latter two factors cancel in the visibilityappraisal, leaving just the luminous intensity of the single marker.

Measure VID1 corresponding to Functional Retroreflectance FRD1 andmeasure VID2 corresponding to Functional Retroreflectance FRD2 aredefined in Table 1. The functional retroreflectance is provided inmillicandelas per lux (mcd/lx) and the distance in metres for thesedefinitions. The constant 1000 in the definition has units(metre^(0.5)·steradian), making VID1 and VID2 unitless. VID1 isdetermined definitely from three R_(I) measurements on a marker, andVID2 is determined definitely from two. No further data are required forthe determination of VID1 or VID2.

Table 1 includes measure VIO1 which is derived from measure VID1 byreplacing the distance term D in the definition of VID1 with thequantity 50 divided by the observation angle. The constant 50 in thedefinition of VIO1 has units (metre·degree). Measure VIO2 is derivedfrom measure VID2 in a corresponding manner.

When using the definitions in Table 1, coefficients of luminousintensity are to be measured in mcd/lx, angles are to be measured indegrees, and distances are to be measured in metres. TABLE 1 Measures ofMarker Performance FRO1 Average of: International Functional R_(I) atObservation Angle = α with Epsilon = −45° Coefficient of Luminous R_(I)at Observation Angle = α with Epsilon = 0° Intensity at ObservationR_(I) at Observation Angle = α with Epsilon = +45° Angle α FRO2 Averageof: Asymmetrical Functional R_(I) at Observation Angle = 1.2α withEpsilon = −50° Coefficient of Luminous R_(I) at Observation Angle = 0.8αwith Epsilon = +20° Intensity at Observation Angle α FRD1 Average of:International Functional R_(I) at Observation Angle = 50/D with Epsilon= −45° Coefficient of Luminous R_(I) at Observation Angle = 50/D withEpsilon = 0° Intensity at Distance D R_(I) at Observation Angle = 50/Dwith Epsilon = +45° FRD2 Average of: Asymmetrical Functional R_(I) atObservation Angle = 60/D with Epsilon = −50° Coefficient of LuminousR_(I) at Observation Angle = 40/D with Epsilon = +20° Intensity atDistance D VID1 1000 × FRD1/D^(2.5) International Visibility Index atDistance D VID2 1000 × FRD2/D^(2.5) Asymmetrical Visibility Index atDistance D VIO1 1000 × FRO1 × (α/50)^(2.5) International VisibilityIndex at Observation Angle α VIO2 1000 × FRO2 × (α/50)^(2.5)Asymmetrical Visibility Index at Observation Angle α

The pair of Table 1 definitions FRO1 and FRD1, imply a correspondence ofobservation angle 50/D with distance D. The pair of Table 1 definitionsFRO2 and FRD2 imply a correspondence of two observation angles 40/D and60/D with distance D. It is helpful to regard the average of these twoobservation angles as a single “vehicular observation angle”. Also thesingle observation angle 50/D in the first pair may be regarded as a“vehicular observation angle”. The correspondence of observation angleswith distance will be understood as a correspondence of vehicularobservation angles with distance in the following paragraphs.

Suppressed or limited long-distance visibility for a retroreflectivearticle, such as a raised pavement marker, is obtained byretroreflective tuning. Three key distances in the road scenario areidentified: “too far”; “far enough”; and “close enough”. The article maybe made so that its functional coefficient of luminous intensity is lowfor observation angles less than a first selected observation anglecorresponding to “too far”, yet high enough for adequate visibility at asecond observation angle, corresponding to “far enough”, and thencontinuing so it still has no less than a third value at a thirdselected observation angle, corresponding to “close enough”, so that themarker is highly visible between the distances corresponding to thesecond and third observation angles. The first selected observationangle is that associated with a distance from the marker correspondingto that point beyond which a driver of a vehicle would have excessiveroad preview with potential distraction of the driver. For example, thisobservation angle may about 0.2 degrees, corresponding to about 250metres road distance for a driver of an oversized passenger vehicle. Thesecond selected observation angle may be about 0.4 degrees,corresponding to about 125 metres for a driver of an oversized passengervehicle. The third selected observation angle corresponds to thatdistance near to the marker at which the marker ceases to need to behighly visible to the driver. This observation angle may be about 1.0degrees, approximately corresponding to about 50 metres road distancefor the driver of a oversize passenger vehicle. Any other observationangles may also be chosen for the first, second, or third observationangle.

Currently, ASTM D4280-04 (Standard Specification for Extended Life Type,Nonplowable, Raised Retroreflective Pavement Markers) provides thatwhite raised retroreflective pavement markers shall have an R_(I) of atleast 279 millicandelas per lux at an observation angle of 0.2 degrees,Epsilon angle of 0 degrees, and a horizontal entrance angle of 0degrees. Most standard markers manufactured to this specification havemuch higher R_(I) than this at these angles, because road markers mustbe made with allowance for wear. These standard markers have highvisibility at long distance until they are well worn.

Excessive road preview is a function of the intensity of the lightreaching the driver from very far away markers. When the VID1 or theVID2 of the marker is less than a first value, the visibility to thevehicle driver is reduced such that the driver does not have excessiveroad preview, thereby avoiding distraction of the driver.

The maximum VID1 and VID2 values established for excessive road previewand driver distraction may have different values depending on thespecific circumstances of the road, traffic, climate, etc.Correspondingly, the values for the functional coefficient of luminousintensity and the first, second, and third observation angles will varydepending upon the specific application. These values may also beslightly different for white, yellow, or other colored markers.

While the embodiment described herein relates to the use of a whiteraised pavement marker, the invention includes raised pavement markersof any color, and includes cube corner elements of any color. Thisincludes white, yellow, red, green, and blue, as set forth in ASTMD4280-04, as well as any other color. Color may be selected withoutdeparting from the spirit and scope of the invention.

The R_(I) values discussed herein apply to standard sized road markershaving approximately 10 cm² of lens area, measured projected in theforward direction. Retroreflectance scales with lens area, so if acertain R_(I) value is achieved by markers discussed herein, markershaving lens area X, in square centimetres, measured projected in theforward direction, would have X/10 times the mentioned R_(I) value.Similarly, the values of the functional coefficients of retroreflectionand the values the visibility indices scale with area.

For some markers of the present invention, the value of the functionalcoefficient of luminous intensity follows a curve having a single peaknear the “far enough” observation angle. The functional coefficient ofluminous intensity for a marker made in accordance with the presentinvention in FIGS. 14A and 14C exemplifies this characteristic.

In one embodiment, the maximum value of the functional coefficient ofluminous intensity FRO1 occurs between observation angles of about 0.3degrees and about 0.4 degrees. The functional coefficient of luminousintensity is less than the maximum value at observation angles less thanabout 0.3 degrees and greater than observation angles of about 0.4degrees. In one embodiment, the functional coefficient of luminousintensity at an observation angle of about 0.4 degrees is greater thanthe functional coefficient of luminous intensity at observation anglesof about 0.2 degrees or less. See FIG. 14A.

In one embodiment, the maximum value of the functional coefficient ofluminous intensity FRO2 occurs between observation angles of about 0.4degrees and about 0.5 degrees. The functional coefficient of luminousintensity is less than the maximum value at observation angles less thanabout 0.4 degrees and greater than observation angles of about 0.5degrees. In a preferred embodiment, the functional coefficient ofluminous intensity at an observation angle of about 0.5 degrees isgreater than the functional coefficient of luminous intensity atobservation angles of about 0.2 degrees or less. See FIG. 14C.

In contrast, prior art pavement markers are designed to maximize thefunctional coefficient of luminous intensity at small observation anglesso as to increase the visibility of the marker at long distances. See,e.g., the functional coefficient of luminous intensity for the standardmarker in FIGS. 14A and 14C. The intensity generally is a rapidlydeclining function of the observation angle—the intensity is greatest atthe smallest observation angles and decreases rapidly as the observationangle increases. For example, the functional coefficient of luminousintensity for the standard marker at an observation angle of about 0.4degrees is less than one-third the functional coefficient of luminousintensity at an observation angle of about 0.2 degrees. See Tables 3 and4, below.

The Visibility Index of a retroreflector in accordance with the presentinvention is less than 0.5 at observation angles of about 0.2 degreesand less, corresponding to a distance of about 250 metres and more, asshown in Tables 5 and 6, below. The Visibility Index of a retroreflectorin accordance with the present invention is also less than that of thestandard marker for observation angles of about 0.3 degrees and less,corresponding to a distance of about 167 metres, and greater than thatof the standard marker for observation angles of at least between about0.4 and about 1.0 degrees, corresponding to a distance of between about125 metres and about 50 metres, as shown in Tables 5 and 6, below.

The retroreflective article of the present invention preferably includesa layer of optically clear material having a smooth front surface and aplurality of retroreflective cube corner elements on the reversesurface. The cube corner elements are preferably macro cube cornerprisms with diameters between about 1 mm and about 3 mm. The materialmay be any conventional plastic used for retroreflective items,including polycarbonate, vinyl, nylon, methyl methacrylate, and opticalgrade acrylonitrile butadiene styrene. The material is preferably athermoplastic resin, for example, polycarbonate. Other opticalmaterials, such as glass, may also be used. Material considerationsinclude parameters other than optical qualities, such as hardiness ordurability in the specific application, that will influence the choiceof materials.

FIG. 1 illustrates a retroreflective article in accordance with thepresent invention, in which a raised pavement marker 10 has a housing 12and a retroreflective lens 14. The housing 12 may be of any conventionalmaterial suitable for use as a housing, such asacrylic-styrene-acrylonitrile, and is preferably a thermoplasticmaterial. The lens 14 may be of any transparent optical material such asacrylic, polycarbonate, or nylon and has a substantially smooth frontface 18 for receiving light and a rear face 20 having a plurality ofretroreflective cube corner elements 22 thereon. The angle of the lensassembly 14 with respect to the horizontal is preferably about 35degrees, but may be any angle. Light rays, such as p, substantiallyparallel to the pavement surface 24, are refracted at the lens frontface 18 to enter the cube corners 22 as light rays q.

FIG. 2 is a geometrical representation of a cube corner element 22, withthree mutually perpendicular faces 26, 28, and 30. The square shapes ofthe illustrated faces 26, 28, and 30 are not necessary for a cubecorner. An incident light ray meets and is refracted by the frontsurface 18, illustrated in FIG. 1. For retroreflection, the refractedray meets any face of the cube corner element 22, is reflected, meets asecond face, is reflected, meets the third face, and is reflected toagain meet the front surface 18 where it is again refracted. FIG. 2shows the cube axis 29 having a direction that makes equal angles to thethree faces 26, 28, and 30. It is common for cube corner elements to beconfigured within a tilted road marker lens in such manner that thedirection q is parallel to the axis of the cube corner. Cube corners soconfigured in a lens are called “axial”.

FIG. 2 illustrates that each cube corner element 22 has three dihedraledges 32, 34, 36. FIGS. 4A-D are drawn in the view in the refracteddirection q from FIG. 1. One dihedral edge may be located so as toappear vertical in FIGS. 4A and 4B, or horizontal as in FIGS. 4C and 4D.In each case, the identified dihedral edge corresponds with dihedralerror e₃. FIG. 3 illustrates the dihedral angle 40 between faces 26 and28 viewed in the direction of dihedral edge 32. In a perfect cube cornerelement 22, the angle between each pair of faces 26, 28, and 30 at eachof the dihedral edges 32, 34, and 36 is a 90 degree angle, as indicatedby right angle 38 in FIG. 3. The actual angle 40 illustrated in FIG. 3is larger than the right angle 38. The actual angle 40 may also be equalto or smaller than the right angle 38. The numerical difference betweenthe actual angle 40 and a right angle 38 at dihedral edge 32 is thedihedral angle error for edge 32. The dihedral angle error for edge 32is positive when angle 40 measures greater than 90 degrees and isnegative when angle 40 measures less than 90 degrees. In making a cubecorner tool it is possible to modify dihedral angle 40 to have anydesired error. Independently, the angle at each of the other dihedraledges 34 and 36 may be modified from a right angle to any larger orsmaller angle. The three dihedral angle errors characterizing a cubecorner determine the directions of reflected light relative to thedirection of incident light, for each of the six scenarios based on theorder of the faces encountered, as explained in Yoder's cited paper.

The retroreflectance of such cube corner elements 22 placed into, forexample, a raised pavement marker is affected by the angle of the lensassembly 14 with respect to the horizontal. In a preferred embodiment,the angle of the lens assembly 14 with respect to the horizontal isabout 35 degrees, but any practical angle may be used, such as 30degrees, 45 degrees, or any other angle greater than zero degrees andtypically less than 90 degrees. The pattern of retroreflection will beinfluenced by this angle, because there is refraction at the front face18.

FIG. 5 illustrates an idealized reflection spot pattern of a typicalcube corner element of the kinds illustrated in FIGS. 4A and 4B andhaving three equal dihedral angle errors in a tilted marker lens. Twospots 46, two spots 48, and two spots 50 represent the pattern ofretroreflection. With respect to a raised pavement marker, upper spots46 represent the light that is reflected for visibility at observationangles greater than a predetermined observation angle, such as forvisibility by a vehicle driver, lower spots 48 represent light that isreflected toward the pavement, and middle spots 50 represent lightreflected that also does not reach a vehicle driver to contribute tovisibility. Note that only 2/6 of the light, that represented by spots46, is reflected for visibility.

FIG. 6 illustrates an idealized reflection spot pattern of a cube cornerelement with three equal dihedral angle errors in which the element isrotated 90 degrees around its axis in a tilted marker lens. Suchelements are illustrated in FIGS. 4C and 4D. The reflected lightrepresented by three top spots 52 is reflected for visibility by avehicle driver at observation angles greater than a predeterminedobservation angle and three bottom spots 54 represent light reflectedtoward the pavement. FIG. 7 illustrates a similar reflection spotpattern in which three top spots 52 are adjusted to provide a moreconcentrated reflection pattern by selection of dihedral angle errors.FIGS. 6 and 7 illustrate that 3/6 of the light is usefully reflected,instead of 2/6 of the light, as shown in FIG. 5.

FIG. 8 illustrates a spot pattern achieved using cube corners of thekinds illustrated in FIGS. 4A and 4B and in which three top spots 52 arecoincident. The spot patterns of FIGS. 7 and 8 are achieved bymanipulating the dihedral angle errors of the cube corner elements inaccordance with the present invention. All spot patterns from plasticretroreflectors must be understood as idealized spot patterns. Theactual spot patterns are not so clear as illustrated. The idealizedpatterns are the underlying patterns which randomization and distortionsmay obscure.

A cube corner element of the kind illustrated in FIGS. 4A and 4B havingdihedral angle errors e₁, e₂, and e₃ of about 0 degrees, −0.13 degrees,and 0 degrees, respectively, will exhibit an idealized pattern of spotsillustrated in FIG. 8 with a front face 18 at a 30 degree angle withrespect to the horizontal. Raised pavement markers having about half ofthe cube corner elements with the −0.13 degree dihedral angle error one₂ and half having the −0.13 degree dihedral angle error on e₃ willexhibit an idealized spot pattern combining FIG. 8 and the mirror imageof FIG. 8 around a vertical axis to achieve the spot pattern shown inFIG. 9D that limits the range of the retroreflector.

The present invention results in a reflected idealized spot pattern inwhich the spots are directed mostly to the top of the reflectionpattern, as illustrated in FIGS. 7 and 8. This selectively limits therange of visibility of the retroreflector.

The dihedral angle errors in the cube corners of a cube corner tooldetermine a pattern of spots. The cube corner tool is described in moredetail below. If a cube corner of the tool is aligned as in FIG. 4A or4B, and if it has the three dihedral angle errors {e₁, e₂, e₃}, Yoder'scited paper explains how the six spots then have locations inrectangular coordinates given by the algebraic expressions in Table 2.The dimensions of the coordinates, e.g., degrees, is the same as thedimensions of the dihedral angle errors. The algebraic expressions givevery good approximations to the exact trigonometric expressions providedthe incident light reaches the cube corners axially, as is the case withtypical road marker lenses when the horizontal entrance angle is zerodegrees. The spots are assumed to be viewed from the direction of thelight entering the cube corner. TABLE 2 Spot Coordinates x y Spot 1{square root over (⅔ (e₁ − e₂ − 2e₃) {square root over (2)} (−e₁ − e₂)Spot 2 {square root over (⅔ (e₁ + e₂ + 2e₃) {square root over (2)}(−e₁ + e₂) Spot 3 {square root over (⅔ (−e₁ + e₂ − 2e₃) {square rootover (2)} (e₁ + e₂) Spot 1′ {square root over (⅔ (−e₁ + e₂ + 2e₃){square root over (2)} (e₁ + e₂) Spot 2′ {square root over (⅔ (−e₁ − e₂− 2e₃) {square root over (2)} (e₁ − e₂) Spot 3′ {square root over (⅔ (e₁− e₂ + 2e₃) {square root over (2)} (−e₁ − e₂)

For cube corners that are aligned as in FIG. 4C or 4D, the x and ydimensions in Table 2 are interchanged.

Table 2 may be used to determine the spot patterns from tools. In orderto use Table 2 to determine the idealized spot pattern from plastic roadmarker lenses formed from the tools, first the values e₁, e₂, and e₃need to be altered according to the transformation of the manufacturingprocess, discussed in greater detail below. Second, the formulas areapplied. Third, the resulting values of x and y are multiplied byfactors which depend on the refractive index of the plastic and theslope of the marker lens face. Refraction at the sloping lens frontsurface stretches or elongates, more vertically than horizontally, thespot pattern. The x values from Table 2 must be multiplied by n, therefractive index of the plastic, and the y values from Table 2 must bemultiplied by $\frac{\sqrt{n^{2} - {\cos^{2}\theta}}}{\sin\quad\theta}$where θ is the angle between the marker face 18 and the ground 24.

Dihedrally aberrated cube corners are specified by three parameters, e₁,e₂, and e₃. The variety of spot patterns must therefore be a threeparameter family, or a small number of three parameter families. Mostpatterns are not achievable by reflection from dihedrally aberrated cubecorners. Table 2 shows how the patterns of reflected spots areconstrained. Spots 1′, 2′, 3′ are just the antipodes of spots 1, 2, 3respectively, implying that all achievable patterns have centralsymmetry. The triplet of spots 1, 2, 3 is further constrained by thealgebraic relations in Table 2.

In particular, it can be shown from Table 2 that if any pattern of sixspots is achievable by reflection from dihedrally aberrated cube cornersthen that pattern is achievable in only two ways, namely by someparticular {e₁, e₂, e₃}, and by the negative aberration, {−e₁, −e₂,−e₃}. This invention realizes useful spot patterns achieved bydihedrally aberrated cube corners.

For the present invention, for cubes oriented as in FIG. 4A or 4B, aspot pattern is desired where y coordinates are all far from zero.Examining the y expressions in Table 2, this requires that the absolutevalues of e₁ and e₂ must be very different. It is also desired that nospot have x coordinate far from zero. Examining the x expressions inTable 2, the first two terms in parentheses include all theparenthesized parts of the y expressions. Since these, when multipliedby √{square root over (2)}, are already far from zero, and since therewill be 2e₃ both added to and subtracted from some of them in the xexpressions, the only way to not have large x coordinates is to make theabsolute value of e₃ small. A simple solution will have the absolutevalues of e₃ and one of e₁ or e₂ small, such as less than about 0.03°,while the absolute value of the other of e₁ or e₂ is large, such asgreater than about 0.10°. Thus an aberration like {0°,0.13°, 0°} issuggested by the desired pattern and the expressions of Table 2.

Cube types that have been used or proposed for road marker lenses fallinto two kinds: a first kind with a dihedral edge in a left-rightsymmetry plane, and those rotated 90° from the first kind. For cubecorners of the second kind, the x and y dimensions in Table 2 areinterchanged. Inspection of Table 2 shows that no spot patternsachievable by dihedral errors of cubes of the first kind are alsoachievable by dihedral errors of cubes of the second kind, and viceversa.

FIGS. 4A and 4B illustrate schematically the kind of cube corner thathas a left-right symmetry plane. One dihedral edge must be in theleft-right symmetry plane. The view is in direction q shown in FIG. 1.FIGS. 4C and 4D illustrate schematically a second kind of cube cornerwhich would be of the first kind except for a rotation of 90° about thedirection q and viewed in the direction q. There is a dihedral edgeappearing horizontal in each of FIGS. 4C and D, but this edge is notnecessarily horizontal in the lens. The cube shapes in FIGS. 4A-D areanonymous, and not pertinent to the distinction into two kinds. Thus,the cube shapes may be triangular, rectangular, hexagonal, or any othersuitable shape without departing from the spirit or scope of theinvention. FIGS. 4A-D also indicate the convention for denoting thedihedral angles according to the error terms e₁, e₂, and e₃.

In this application, the road marker examples have their lens assembly14 at 30 degrees with respect to the horizontal and made ofsubstantially acrylic plastic having a refractive index of 1.49. Themethods of achieving limited range are equally applicable to other lensslopes and other refractive indices. The two stretch factors, n and$\frac{\sqrt{n^{2} - {\cos^{2}\theta}}}{\sin\quad\theta},$adjust the results of Table 2 to various slopes and indices. For usewith non-axial cube corners it is suggested that the axial case becalculated as first approximations and final verification be done by raytracing. For designing light patterns at large horizontal entranceangles, ray tracing is necessary. Metallization does not directly changethe effects of the dihedral angle errors for macro sized cube corners,but it does for micro sized cube corners, by its effect on diffraction.For both, the reduced overall retroreflectance of the metallized markermay necessitate modification of the dihedral angle errors to effect thedesired spot pattern and visibility characteristics.

FIG. 18 illustrates how a spot pattern may change for horizontalentrance angles very different from zero degrees. The cube corners forthe illustration are acrylic of the kind in FIG. 4A with dihedral errors{0°,−0.13°, 0°} configured axially in a road marker having its lens facetilted 30° with respect to the road surface. The spot pattern continuesto consist of two spots but the pattern changes size and rotates withchange of horizontal entrance angle. The spot denoted “0” and itsantipode in FIG. 18 correspond to zero degree horizontal entrance angle,and are the same as in FIGS. 8 and 9C. The spot denoted “+20” and itsantipode correspond to +20° horizontal entrance angle and the spotdenoted “−20” and its antipode correspond to −20° horizontal entranceangle. Positive horizontal entrance angle pertains to markers on roadscurving to the left; negative horizontal horizontal entrance anglepertains to markers on roads curving to the right. Twenty degrees is thegreatest horizontal entrance angles generally required from roadmarkers. The pattern distortion illustrated in FIG. 18 is of similarkind and degree to that of other spot patterns at the these entranceangles. An idealized spot pattern that is highly efficient at zerodegree entrance angle will generally be acceptably efficient at thelarger entrance angles. Nevertheless raytrace analysis or experimentalconfirmation is recommended.

If a color other than white is selected for the cube corner elements, orfor the raised pavement marker, then the dihedral angle errors may bedifferent to achieve the desired limited long-distance visibility.Generally, use of a color other than white will result in a reducedfunctional coefficient of luminous intensity achieved by the marker.Choice of color may not directly change the effects of the dihedralangle errors, but the choice of color may require changing the dihedralangle errors to increase the functional coefficient of luminousintensity achieved by the marker. Or the target functional coefficientsof luminous intensity to be achieved by the marker may be different fora colored marker than for a white marker. These specific dihedral angleerrors may be determined without undue experimentation in the event thata color other than white is selected.

The cube corner elements are manufactured, for example, by conventionalprecision injection molding. One way in which to manufacture such cubecorner elements 22 is generally to make a master plate by clusteringmetal pins having male cube corners ground and polished onto their tipsor otherwise creating a pattern of male cube corners on a planar surfaceof a master plate. The master plate is then used to create one or moretools, comprising female cube corners, such as by electroforming.Further generations of male, female, etc. may be created, such as byelectroforming. The final working tool is a mold comprising female cubecorners into which the transparent lens material is placed. The lensmaterial is allowed to take a shape corresponding to the cube corners ofthe mold. The lens material is allowed to harden and is removed from themold. The lens material may be further cured either while in the mold orafter removal from the mold to achieve the desired degree of hardnessand other properties.

The manufacture of such tool is known in the art and, except asdiscussed herein, any conventional manner of making such a tool may beutilized without departing from the spirit and scope of the presentinvention. The mold may be of any conventional material suitable for useas a mold, preferably electroform nickel.

While the faces of the tool's cube corners may be extremely flat, thefaces of molded cube corners are generally much less flat. Dihedralangles could properly be assigned to the intersections of the threeplanes that best fit the cube faces. The most convenient way to measurethe dihedral angles of a flat faced female cube corner is with aninterference microscope. The interference microscope measures the slopesof six portions of the wavefront leaving the cube corner. When the cubecorner's faces have curvature, the six portions of the wavefront havecurvature. The curvatures of the individual faces cannot beunambiguously determined from the curvatures of the wavefront. Fittingsix best fitting planes to the six portions of the wavefront istherefore not equivalent to fitting three planes to the three cubefaces. The dihedral angles of the molded macrocube corners must beunderstood as unmeasured idealizations, just as the spot patterns frommolded macrocube corners are idealizations. The magnitude of thedihedral angle errors in the cube corner element 22 may be estimated.

The difference between the working tool dihedral angle errors and theresulting cube corner element 22 dihedral angle errors may be used topredict the resulting cube corner element 22 dihedral angle errors basedon the tool dihedral angle errors, adjusted for the estimated errorshrinkage or growth by the particular lens forming process used. Thisallows for determination of the proper tool dihedral angle errors toachieve the desired cube corner element 22 dihedral angle errors. Forexample, if the particular process used shrinks all dihedral angleerrors by 0.05 degrees, and if the desired lens cube corner element 22dihedral angle errors are 0 degrees, −0.13 degrees, and 0 degrees, thenthe tool dihedral angle errors are made to be +0.05 degrees, −0.08degrees, and +0.05 degrees. Use of such a tool with this processresulting in the known dihedral angle error shrinkage then yields thedesired cube corner element 22 dihedral angle errors.

The male cube corner prisms illustrated as elements 22 in FIG. 1 aremolded or otherwise formed from generally nickel metal havingcorresponding hollow female cube corners. The tools are generallyevolved from machined masters by nickel electroforming. The machinedmasters themselves may be of many materials, even plastic materialsprovided their surface can be made conductive for the electroforming.The evolution from master to final tool may involve severalelectroforming steps, and these may modify the dihedral angles of thecube corners. The stress of the deposited nickel playing upon thegeometry of the arrayed cube corners can produce unequal deviations ine₁, e₂, and e₃ during the evolution of the tool. However, the fact thatrather unusual values of e₁, e₂, and e₃ may be required for execution ofthis invention introduces no additional complication. Whatever angleadditions or subtractions are found to occur to e₁, or to e₂, or to e₃during the evolution of a prior art tool having the same form of arrayof cube corners will occur also with the respective e₁, e₂, and e₃ ofthe present invention.

The rather unusual values of e₁, e₂, and e₃ in tools which may berequired for the present invention will generally require rather unusualvalues of e₁, e₂, and e₃ in the machined masters from which the toolsevolve. To make pins with general dihedral angles e₁, e₂, and e₃requires setting three different tilts for the grinding and polishing ofthe three different faces. In the case of micro cubes, the cited U.S.Pat. No. 4,775,219 describes ruling methods for producing some desireddihedral angles with the complication that these methods also producesome undesired dihedral angles. The methods given in U.S. Pat. No.6,015,214 to Heenan, et al., avoid this problem.

A cube corner element made from a cube corner tool has dihedral angleerrors corresponding to the dihedral angle errors in the cube cornertool. The process of molding the cube corner elements from the cubecorner tool transforms the angles in the cube corner elements in asimple additive way. These addends are denoted T₁, T₂, and T₃,corresponding to the errors e₁, e₂, and e₃, respectively. Injectionmolding typically results in shrinkage of the cube corner which causessubtractions from the dihedral angles. In a flat retroreflectorcomprising many hexagonal cube corners there is perfect 120° rotationalsymmetry, and the dihedral angles generally all transform alike.However, in the tilted retroreflective lens for a road marker wherethere is pronounced asymmetry, the three dihedral angles generallytransform by unlike amounts. The amounts depend not only on the cubegeometry but on the plastic material and the conditions of molding.

The process of molding transforms the dihedral angles {e₁, e₂, e₃} ofthe tool into new dihedral angles {e₁+T₁, e₂+T₂, e₃+T₃} in the plasticlens. The values of the transformation addends depend on the corner cubeshape and size and configuration, the thickness of the reflex lens, itsmaterial, and its forming process. These values may be determined foreach specific application without undue experimentation. For a commonmarker lens designed for 30° slope with hexagonal cube corners havingdiagonals 2.76 mm and oriented as in FIG. 4A, the total lens thicknessbeing about 3.43 mm, the plastic being an acrylic and the formingprocess injection molding, under the particular molding conditions ofthe trial the {T₁, T₂, T₃} was found to be approximately {−0.04°,−0.04°, −0.07°}.

FIGS. 9A-C illustrate how the reflection pattern from the molded part isdetermined by the reflection pattern from the tool. FIG. 9A shows thepattern from a cube of the tool. According to Table 2, this patterncould only be produced by cubes having dihedral errors {e₁,e₂,e₃} equalto{+0.04°, −0.09°, +0.07°} or equal to the negatives {−0.04°, +0.09°,−0.07°}. Suppose that the tool's cubes satisfy the first solution,{+0.04°, −0.09°, +0.07°}. Molding imposes an additive transformation{−0.04°, −0.04°, −0.07°} on the cube tool's dihedral angles, producingthe new dihedral errors {0°, −0.13°, 0°}. The equations of Table 2 givethe spot pattern for {0°, −0.13°, 0°} shown in FIG. 9B as if it werestill a tool. Notice that the spot pattern has compressed into twoapparent spots, each containing three reflected spots. The molded cubecorners being part of a tilted plastic lens, produces a modification ofthis pattern. Refraction at the lens front surface stretches orelongates, more vertically than horizontally, the spot pattern. The xvalues from Table 2 must be multiplied by n, the refractive index of theplastic, and the y values from Table 2 must be multiplied by$\frac{\sqrt{n^{2} - {\cos^{2}\theta}}}{\sin\quad\theta}$where θ is the angle between the marker face 18 and the ground 24. FIG.9C shows the stretched spot pattern from an n=1.49 acrylic lens, havingslope θ=30° to the ground. FIG. 9D shows the doubling of the spotpattern of FIG. 9C as would result by including mirror image cubes inthe tool. The pattern of FIG. 9D is not achievable by means of dihedralerrors on a single cube corner.

FIG. 12 illustrates the light intensity which might be expectedcorresponding to a certain point pattern of a well-molded acrylic roadmarker lens of the prior art. For this illustration, the molded dihedralerrors were {0.04°, 0.04°, 0.04°}. The hexagonal spot pattern is asgiven by the Table 2 expressions and then stretched 1.49×horizontallyand 2.425×vertically to account for refraction at the 30° sloped frontface. Each point is shown spreading into an approximately Gaussianpattern, again with the proportional stretching, characteristic of highquality injection molding. The successive contours around each spotindicate 10%, 20%, etc. of the intensity at the peak.

For most cube designs the six mounds contain equal light. The lightmounds add to a lumpy ring pattern for this retroreflector. A spotpattern is an abstraction from an actual light pattern. If the cubes aremacrocubes, and if they are closely agreeing in their aberrations, andif their faces are highly flat, then the spot pattern is clearly visiblein the light pattern. In other cases it might be possible to infer aspot pattern from the light pattern. In yet other cases it is notpossible to infer the spot pattern from the light pattern. Spot patternsfrom injection molded macrocube retroreflectors must be understood as anidealization.

It is also possible to design cube corner elements in which the quantityof light corresponding to the different spots is definitely unequal.FIG. 17 illustrates the q view of a rectangular cube corner in solidlines. Dashed lines segment the cube corner into the six areas whichcatch light for the corresponding six spots 1-3′, labeled according toTable 2. Spots 2 and 2′ receive a decreasing fraction of the total lightas the rectangle is narrowed. When using such cube corner elements,wherein four of the six spots contain almost the whole energy,application of the present invention is simplified. The positions ofspots 2 and 2′ in the spot pattern have reduced importance. For the cubecorners illustrated in FIG. 17, each of spots 2 and 2′ carryapproximately one-fourth the energy of each of the other four spots.Making the rectangle shape less tall causes each of spots 2 and 2′ tocarry more, for example one-half, the energy of each of the other fourspots.

The descriptions of this invention presume that the molded cubes thatwere intended to be alike have mean dihedral aberration {e₁, e₂, e₃}. Tothis mean dihedral aberration there corresponds a spot pattern. Theactual collection of molded cubes intended to be alike are not perfectlyalike. The actual molded cubes have different e₁, e₂, e₃ from the meanvalues, and each could be said to have a different spot pattern fromthat associated with the mean dihedral aberration. Also the actualmolded cubes may have imperfect face flatness. Even the front surface ofthe lens may have significant local unflatness. These deviations can beunderstood as producing a light mound in place of a spot.

This approach is also applicable to microcubes. With microcubes, themethods of plastic forming, such as embossing and casting, need to bemore accurate than injection molding if only because unsharpness ofedges would result in great loss of optical efficiency. With microcubesthe formed cubes intended to be similarly aberrated are very nearly soand faces intended to be flat are very nearly so. Diffraction is themain source of deviation from the expected spot pattern, and whilediffraction does not function as a similar spreading of each of thespots, unmetallized microcubes with active apertures at least about 200microns in diameter and metallized microcubes with active apertures atleast about 150 microns in diameter exhibit a calculated spot patternthat is an adequate skeletal indicator of the realized light pattern.The active aperture of a cube corner is that portion of the full shapeof the cube corner which, for a particular Beta and Omega,retroreflects. How diffraction phenomena influence the geometricaberrations is well known from the optical arts of image forming lenses.

Microcubes are ill suited for experimentation, but mathematical modelingthat takes into account aberration with diffraction for microcube cornerreflection can be based on the principles contained in Yoder, op cit,and E. R. Peck, “Polarization Properties of Corner Reflectors andCavities,” Journal of the Optical Society of America, Volume 52, Number3 (March 1962). Such modeling illustrates microcubes to be successfulrealizations of the invention, because the diffraction from unmetallizedmicrocubes with active apertures at least about 200 microns in diameterand metallized microcubes with active apertures at least about 150microns in diameter is sufficiently small to allow gradients in thelight pattern about as steep as the best injection molding of macrocubesallows and microcubes this size have no significant diffraction lightpeak at the center of the pattern.

When the molding transform of different parts of the same lens varies,it is useful to study the light pattern from each of the different partsof the lens.

Cube corner elements may also be made by casting, embossing, or othersuitable conventional methods in which the dihedral angle errors of theresulting cube corner elements may be controlled to provide the desiredlimited retroreflectance range.

FIG. 11 illustrates the spots corresponding to passenger car observationin the distance range 40 m to 130 m. For a reflection to reach the cardriver viewing the retroreflector from behind the left headlamp, thereflection must diverge in a direction shown by the swarm of points onthe right side of the Figure. Similarly for the right headlamp and theswarm of points on the left side of the figure. For FIG. 11, sevendifferent passenger car geometries were considered, and road markerswere considered on both sides of the driving lane and one lane fartherleft and right. The asymmetrical shape of FIG. 11 results from thedriver sitting left of center in the car.

The pattern illustrated in FIG. 13A, corresponding to the InternationalFunctional Coefficient of Luminous Intensity and the InternationalVisibility Index, has been proposed to bridge differences between thosecountries where vehicle drivers sit left of center (such as the USA) andcountries where vehicle drivers sit right of center (such as the UK).More than ¼ of the world's population are in countries where vehicledrivers sit right of center (e.g., India, Japan, UK.) FIG. 13A shows thepattern of retroreflection divided into regions. The shaded trapezoidalregion ABEF is where the pattern should concentrate its intensity. Thetriangular region FEO is where the pattern should be weak in order tolimit long distance visibility. Additional triangular regions ODA andOBC should also be weak in order to conserve light for region FEO.

The regions are defined by upper and lower vertical limits. In FIG. 13A,the range is 0.3 deg to 1.0 deg.

The shaded region ABEF is defined by the two inequalities:−45°≦ε≦45°0.3°≦α cos ε≦1.0°This region is designated “

(0.3°;1.0°)”.

The small triangular region is defined by the two inequalities.−45°≦ε≦45°0°≦α cos ε≦0.3°This region is designated “

(0°;0.3°)”

The larger triangle on the left is defined by−90°≦ε≦−45°0°≦α cos ε≦1.0°The larger triangle on the right is defined by−45°≦ε≦90°0°≦α cos ε≦1.0°The union of these two triangular regions is designated “

(1.0°)”.

One result of the present invention is to minimize the R_(I) contentwithin the region

(0°; 0.30) while achieving a large R_(I) content within the region

(0.3°;1.0°), which requires keeping the R_(I) content within region

(1.0°) small. The choice of demarcation at 0.3° is merely exemplary asfalling between a chosen “too far” point and a chosen “far enough”point. The choice of limitation to 1.0° is also merely exemplary of achosen “close enough” point. For the marker of Example 1,

(0°; 0.3°)=45,

(0.3°;1.0°)=219 and

(1.0°)=84

Markers can also be designed with an asymmetrical pattern ofretroreflection for best functioning in one or the other kind ofcountry. For example, FIG. 13B illustrates a desirable visibilityprofile of retroreflected light for countries in which the drivers sitleft of center, as also shown in FIG. 11. The shaded region in FIG. 13Bis in two trapezoidal parts, the left part is defined by the twoinequalities:−65°≦ε≦50°0.3°≦α cos ε≦1.0°

The right part is defined by the two inequalities:−5°≦ε≦45°0.3°≦α cos ε≦1.0°

This two-part region is designated “

(0.3°;1.0°)”.

The region in FIG. 13B corresponding to long distance visibility is intwo triangular parts. They are not indicated with shading in the figure.The left part is defined by the two inequalities:−65°≦ε≦−50°α cos ε≦0.3°

The right part is defined by the two inequalities:−5°≦ε≦45°α cos ε≦0.3°

This two-part region is designated “

(0°;0.3°)”.

One result of the present invention is to minimize the R_(I) contentwithin the region

(0°;0.3°) while achieving a large R_(I) content within the region

(0.3°;1.0°). The choice of demarcation at 0.3° is merely exemplary asfalling between a chosen “too far” point and a chosen “far enough”point. The choice of limitation to 1.0° is also merely exemplary of achosen “close enough” point.

The R_(I) content of a region

of defined α and ε is computed as an integral of the R_(I) over thatregion. Formally, R_(I) content of$= {\underset{R}{\int\int}{R_{I}\left( {\alpha,ɛ} \right)}{\mathbb{d}\alpha}{{\mathbb{d}ɛ}.}}$As this formula is used herein, the angles α and ε are measured indegrees, and the R_(I) is measured at 0 degrees horizontal entranceangle in units of mcd/lx. Because R_(I)(α,ε) is an empirical function,the integration will be a numerical approximation based on measurementsof R_(I) made at many (α,ε) points within the region

. The approximation can be made as exact as desired by increasing thenumber of measurements.

Example 2, below, satisfies the sit-on-left criterion better than theinternational criterion. Mirror image versions of Example 2 wouldcorrespondingly satisfy the sit-on-right criterion.

To make a retroreflector in accordance with the present invention, thedesired light pattern is identified, such as in FIG. 13B. Withmacro-cube corners, an achievable spot pattern, or combination of spotpatterns which approximates the desired light pattern is thendetermined. The process of finding achievable spot patterns requiresexercise of the expressions in Table 2 together with the stretch factorsdescribed above. This can be rendered in a calculating spreadsheet.Alternatively a computer program can be written which finds the nearestachievable spot pattern to any conjectured spot pattern.

With microcubes the spot pattern is only a crude indicator of the lightpattern. Diffraction phenomena can strongly influence the geometricaberrations, as is well known for all optical systems While microcubesare not well suited for experimentation, a calculational approachillustrates that the range of retroreflectance may be limited in themanner described herein. Good diffraction plus aberration mathematicalmodeling of microcube corner reflection is possible using principlescontained in the papers of Yoder and Peck, cited above.

The light pattern from a retroreflector may be measured withconventional retroreflector photometry. The photometer may readintensity at several hundred points, which data may then be assembledinto a picture. The Application angle system described in CIE Pub. 54.2,“Retroreflection: Definition and Measurement”, is practical, and isincorporated herein by reference. First the direction of illumination ischosen, by fixing two angles termed β and ω_(s) in that system. Then thetwo angles termed α and ε in that system are systematically varied, andthe coefficient of retroreflected luminous intensity measured at eachangle combination. A picture is formed in polar coordinates using α asthe radial dimension and ε as the polar angle.

The size of the cube corner elements must be taken into account.Commonly injection molded macro-sized cube corners for road markers haveprojected areas generally in the range 0.6 sq. mm to 12 sq. mm.Micro-sized cube corners for road markers have projected areas generallyless than 0.05 sq. mm. Projected area used to denote the area as viewedstraight towards the road marker as if from a faraway vehicle. The slopeangle θ of the marker face causes the cube corners to look less tallthan they would were they viewed axially from their rears or from withinthe lens. The shortening factor equals$\frac{n\quad\sin\quad\theta}{\sqrt{n^{2} - {\cos^{2}\theta}}}$where n is the refractive index of the lens. FIG. 1 shows this as thedistance between rays q being less than the distance between rays p.

Light diffraction by macro-sized cube corners is too slight to affectretroreflective performance. Thus a geometrical appraisal of theaberrations, including dihedral angle errors, is sufficient. When themacro-cube corner faces are molded to unusual flatness, or when the manymacro-cube corners comprising a retroreflective marker are molded tounusual similarity, the reflected light pattern is six discrete spots.This pattern is functionally undesirable. For such molding conditions,the tool should be made with cubes not all alike. The individual moldedmacro-cube corners will produce six light spots, satisfying the patternefficiency condition, while the sum of the many cube corners' patternswill be a light swath, also satisfying the pattern efficiency condition,as desirable. Execution of the inventive design requires a delicatecontrol of the spottiness of light pattern. Patterns that are tooblurred will not achieve the desired range cutoff. Patterns that are toopointlike will not function for observers. Intentional variety in themacro-cube corners in the tool is the pointillistic solution to thisdilemma.

Likewise when the close distances visibility must be enhanced, two ormore levels of cube aberration will be included in the tool. ExtendingExample 1, below, a first subset of cubes in the tool will have thedescribed errors {+0.04°, −0.09°, +0.07°} and mold to {0.00°, −0.13°,0.00°}; a second subset of cubes in the tool will have errors {+0.04°,−0.22°, +0.07°} and mold to {0.00°, −0.26°, 0.00°}. In this example thesecond subset of cubes has a molded spot pattern which is a two timesexpansion of the first subset of cubes. Accordingly the molded dihedralerrors are doubled. But the tool dihedral errors in the two subsets isnot doubled. This is because the transformation between tool and moldedpart is additive and fixed.

Introduction of dihedral angle error is not the only way to effect lightpatterns satisfying the criteria of this invention. The cube faces canbe curved or faceted. Holographic light diffusers can be incorporatedinto the lens. Various aberrating means can also be used in combination.

EXAMPLE 1

A retroreflective article in accordance with the present invention canbe made with cube corner elements 22, oriented as in FIG. 4A or 4B, thatdirect the retroreflected intensity away from observation angles between0 and 0.2 degrees. In this case, 0.2 degrees is the observation angleselected to represent “too far”; 0.4 degrees is selected to represent“far enough”; 1.0 degree is selected to represent “close enough”. A toolhaving dihedral angle errors of about +0.04 degrees, about −0.09degrees, and about +0.07 degrees can be used to mold acrylic resin in aprocess that transforms dihedral angle errors by about −0.04 degrees,about −0.04 degrees, and about −0.07 degrees, respectively, to producecube corner elements 22 having idealized dihedral angle errors of about0 degrees, about −0.13 degrees, and about 0 degrees. The tool used toproduce this cube corner element has the retroreflected spot pattern ofFIG. 9A. The molded cube corner element in a raised pavement markerhaving a 30 degree sloping front, which is the angle of the lensassembly with respect to the horizontal, has the idealized spot patternof FIG. 9C. The pattern of FIG. 9C avoids reflected light to observationangles between 0 and 0.2 degrees. FIGS. 14A and 14C illustrate thepredicted functional R_(I) values of the raised pavement marker withreflected light from a light source at 0 degrees horizontal entranceangle at various distances from a raised pavement marker made inaccordance with this example. The observation angle is on the x axis andeither FRO1 or FRO2 is on the y axis. The predicted functional R_(I), inmcd/lx, of a marker in accordance with this example of the presentinvention is shown compared with the predicted functional R_(I) of thestandard marker for observation angles up to 1.0 degree.

The predicted functional coefficients of luminous intensity at thevarious observation angles for a raised pavement marker with cube cornerelements made in accordance with this example compared with that of thestandard marker, as illustrated in FIGS. 14A and 14C, are givennumerically in Tables 3 and 4: TABLE 3 International FunctionalCoefficient of Luminous Intensity Comparison for Example 1 FRO1 and FRO1and Observation FRD1 for FRD1 for example 1 vs. angle Distance example 1the standard the standard (degrees) (metres) marker marker marker 0infinity 336 2112 16% 0.1 500 356 1770 20% 0.2 250 407 1207 34% 0.3 167455 710 64% 0.4 125 448 397 113% 0.5 100 381 228 167% 0.6 83 288 147196% 0.7 71 202 101 200% 0.8 63 138 77 179% 0.9 56 96 61 158% 1.0 50 7049 141%

TABLE 4 Asymmetrical Functional Coefficient of Luminous IntensityComparison for Example 1 FRO2 and FRO2 and Observation FRD2 for FRD2 forangle Distance example 1 the standard example 1 vs. the (degrees)(metres) marker marker standard marker 0 infinity 336 2112 16% 0.1 500349 1765 20% 0.2 250 398 1195 33% 0.3 167 482 694 69% 0.4 125 565 383148% 0.5 100 577 217 266% 0.6 83 474 138 344% 0.7 71 337 97 348% 0.8 63215 74 290% 0.9 56 133 59 226% 1.0 50 85 48 179%

At extreme distance, the passenger vehicle driver would see the markerin accordance with this example of the present invention as 16% asbright as the standard marker.

The predicted Visibility Indices at the distances for a raised pavementmarker with cube corner elements made in accordance with this examplecompared with that of the standard marker, as illustrated in FIGS. 14Band D are given numerically in Tables 5 and 6: TABLE 5 InternationalVisibility Index Comparison for Example 1 VIO1 and VIO1 and ObservationVID1 for VID1 for angle Distance example 1 the standard example 1 vs.the (degrees) (metres) marker marker standard marker 0 infinity 0.000.00 0.1 500 0.06 0.32 20% 0.2 250 0.41 1.22 34% 0.3 167 1.27 1.98 64%0.4 125 2.57 2.27 113% 0.5 100 3.81 2.28 167% 0.6 83 4.54 2.31 196% 0.771 4.69 2.35 200% 0.8 63 4.47 2.49 179% 0.9 56 4.18 2.65 158% 1.0 503.93 2.79 141%

TABLE 6 Asymmetrical Visibility Index Comparison for Example 1 VIO2 andVIO2 and Observation VID2 for VID2 for angle Distance example 1 thestandard example 1 vs. the (degrees) (metres) marker marker standardmarker 0 infinity 0.00 0.00 0.1 500 0.06 0.32 20% 0.2 250 0.40 1.21 33%0.3 167 1.34 1.94 69% 0.4 125 3.23 2.19 148% 0.5 100 5.77 2.17 266% 0.683 7.48 2.18 344% 0.7 71 7.82 2.25 348% 0.8 63 6.97 2.40 290% 0.9 565.77 2.55 226% 1.0 50 4.81 2.69 179%

The FRO2 of the marker of this example is no more than about 400 mcd/lxat observation angles of about 0.2 degrees or less and is at least about550 mcd/lx at observation angle 0.4 degrees and continuing to be aboveabout 85 mcd/lx to about 1.0 degrees. The FRO2 of the marker inaccordance with this example is significantly less than that of thestandard marker for observation angles less than about 0.27 degrees andsignificantly greater than that of the standard marker for observationangles between about 0.43 degrees about 1.0 degrees. By “significantly”is meant by a factor of at least 1.8.

The R_(I) content of region

(0°;0.2°) for the marker of this example equals just 17. For comparison,the R_(I) content of region

(0°;0.2°) for the standard marker equals 55. The R_(I) content of region

(0.4°;1.0°) for the marker of this example equals 154. For comparison,the R_(I) content of region

(0.4°;1.0°) for the standard marker equals only 60. The standard markerputs approximately equal amounts of light into the undesired and thedesired regions, while the inventive marker of this example putsapproximately 9 times as much light into the desired region as theundesired region. For these calculations of R_(I) content, 120measurement points were taken within the region

(0°;0.2°) and 1979 measurement points were taken within the region

(0.4°;1.0°).

EXAMPLE 2

This example will satisfy the driver-on-left criterion instead of theinternational criterion. The molded cubes of Example 1 having dihedralangle errors {0, −0.13, 0} are called A-cubes. Molded cubes havingdihedral angle errors {−0.13, 0, +0.13} are called B-cubes. Molded cubeshaving dihedral angle errors {+0.13, 0, +0.13} are called C-cubes. FIG.9C shows the spot patterns from the A-cubes. FIG. 15A shows the spotpattern from the B-cubes. FIG. 15B shows the spot pattern from theC-cubes. The B-cube puts two spots where the C-cube corner puts one, andvice versa. The B-cube pattern and C-cube pattern partly coincide withthe pattern from the A-cubes.

If the molding transformation is {−0.04, −0.04, −0.07} then in the tool,A-cubes have dihedral angle errors {+0.04, −0.09, +0.07}, B-cubes havedihedral angle errors {−0.09, +0.04, +0.20} and C-cubes have dihedralangle errors {+0.17, +0.04, +0.20}.

Either A-cubes , B-cubes, or C-cubes will satisfy the driver-on-leftcriterion. The A-cubes will reflect the vehicle's left headlamp to adriver, but their reflection of the right headlamp will not be observed.The B-cubes will reflect the vehicle's left and right headlamps to thedriver, but with reduced efficiency. That is, in FIG. 15A, whenreflecting the left headlamp, the two spots in the upper right quadrantwill return to the observer, but the others will not, and whenreflecting the right headlamp the one spot in the upper left quadrantwill return to the observer, but the others will not. FIG. 15B favorsthe right headlight.

This exemplifies how every known retroreflector suffers twoinefficiencies, each of a factor of two. First the symmetry of the lightpattern results in half the reflected light aiming below the headlight,where it certainly won't be observed. Second it is impossible to tailora retroreflector to produce more intensity from the two vehicleheadlights than from just one. If the retroflector efficiently reflectslight from one headlight to the observer, then it fails to reflect theother headlight to the observer.

Relying wholly on the left headlight, a road marker lens of A-cubesproduces as great intensity for the left-of-center driver as lenses withB-cubes or C-cubes. However this defeats the redundancy of the vehicle'stwo headlights. Comparison of FIG. 11 with FIGS. 15A-B and FIG. 12 showsthat it is difficult to achieve efficient reflection of the rightheadlight to the left-of-center driver. This is because the rotationangle shown to the left in FIG. 11 is large and the mounding expectedfrom molding in FIG. 12 aligns vertically, rather than at the strongslant. For this reason a lens of B-cubes might be slightly less intenseto the left-of-center driver than a lens of A-cubes, and a lens ofC-cubes of further reduced intensity. Lenses comprising a mixture ofA-cubes and B-cubes will have behavior midway between them. Lensescomprising a mixture of B-cubes and C-cubes will have behavior midwaybetween them.

It is surprising that the left headlamp does not necessarily provide themajor part of the retoreflected intensity. Such is generally the casefor lenses where the retroflectance is a falling function of observationangle, but not for some of the range limited markers of the presentinvention and the spot patterns therefrom. It is also surprising that,in the devices of the present invention, retroreflectance is more nearlyvarying in rectangular coordinates x,y, which are horizontal andvertical components of observation angle. This is because the cubecorners spot pattern is not especially radial and the refractive stretchis mostly in the y direction. Conventionally, it is considered thatretroreflectance is a polar quantity, varying especially withobservation angle, the radial direction from the center of FIGS. 9 and15.

EXAMPLE 3

This example has an extended retroreflection pattern, by means of mixingtwo spot patterns of different scales. FIG. 10 shows the addition of thespot pattern, shown with solid dots, corresponding to dihedral errors{−0.02°, 0.12°, 0.04°} to the spot pattern, shown with hollow dots,corresponding to dihedral errors {−0.03°, 0.18°, 0.06°}. The secondpattern is exactly 1.5×the size of the first. Note that the cubes in thetool used to make these two patterns will not themselves exhibitpatterns differing in scale. For example, if the molding transformationis {−0.04°, −0.04°, −0.07°}, then the cubes in the tools will be {0.02°,0.16°, 0.11°} and {0.01°, 0.22°, 0.13°} which are not proportionate.

Combination of patterns of different size accommodates the variationwith road distance of the positions of the of the observer relative tothe headlights as shown in FIG. 11. A lens comprised of macrocubes willhave between about 100 and about 300 cube corners, allowing muchflexibility of mixtures of aberrated types.

Supposing that equal numbers of the two cube types were used in thisexample marker, The R_(I) content of region

(0°;0.3°) for the marker of this example equals just 33. For comparison,the R_(I) content of region

(0°;0.3°) for the standard marker equals 82. The R_(I) content of region

(0.3°;1.0°) for the marker of this example equals 175. For comparison,the R_(I) content of region

0.3°;1.0°) for the standard marker equals only 82. The standard markerputs equal amounts of light into the undesired the desired regions,while the inventive marker of this example puts approximately 5.3 timesas much light into the desired region as the undesired region. Note thatthis comparison differs from that described for Example 1 because thereis no observation angle buffer between the desired and undesiredregions.

FIG. 16 graphs the VID2 for the Example 3 marker versus the standardmarker.

EXAMPLE 4

This example differs from the first three in pertaining to cube cornersof the second kind, represented in FIGS. 4C and 4D.

A retroreflective article in accordance with the present invention canbe made with cube corner elements 22 that direct the retroreflectedintensity away from observation angles between 0 and 0.3 degrees. Inthis case, 0.3 degrees is the selected first observation angle. The cubecorner element dihedral angle errors would be about +0.04 degrees, about+0.04 degrees, and about +0.09 degrees.

It should not be expected that the dihedral angle shrinkages for thisstructure would be similar to those of Examples 2 or 3. The moldingtransformation must always be determined by experimentation.

In this example, about half of the cube corners are to be as shown inFIG. 4C, and half as shown in FIG. 4D. With rectangular cube cornersthis can be neatly achieved with contiguous pairs. With hexagonal cubecorners the left half the marker lens can be FIG. 4C cubes and the righthalf of the marker lens FIG. 4D cubes, or vice versa. The two parts ofFIG. 18 illustrate assemblies of hexagonal pins for the right half andthe left half of the marker lens. Use of this cube corner element in araised pavement marker having a 30 degree sloping front, which is theangle of the lens assembly with respect to the horizontal, will resultin directing the retroreflected light away from the observation anglesbetween 0 and 0.3 degrees in an idealized spot pattern similar to thatof FIG. 7. A retroreflective article with cube corner elements 22 madein accordance with this example results in utilization of 3/6 of theretroreflected energy. If the cube corners of this example are to bemade of acrylic and non-metallized then it is necessary that they bespecially canted in order that both the FIG. 4A elements and the FIG. 4Belements achieve Total Internal Reflection for retroreflection to bothplus and minus 20° horizontal entrance angle. For example, the pins canbe assembled with their own edges in the direction q of FIG. 1, as isstandard practice, but the cube corners on the pins would be tipped insuch a way that dihedral edge for e₃ (in FIG. 4C or 4D) makes an angleto q which is larger than the angle that the dihedral edge for e₁ makesto q and the angle that the dihedral edge for e₂ makes to q. The amountof tipping necessary is approximately 1 deg. Use of higher refractiveindex material, such as polycarbonate, makes this special cantingunnecessary.

While the present invention has been illustrated by the abovedescription of embodiments, and while the embodiments have beendescribed in some detail, it is not the intention of the applicants torestrict or in any way limit the scope of the invention to such detail.Additional advantages and modifications will readily appear to thoseskilled in the art. Therefore, the invention in its broader aspects isnot limited to the specific details, representative apparatus andmethods, and illustrative examples shown and described. Accordingly,departures may be made from such details without departing from thespirit or scope of the applicants' general or inventive concept.

1-19. (canceled)
 20. A retroreflective lens comprising cube corners,wherein the pattern of retroreflection is such that the R_(I) content ofthe region

(0.4°;1.0°) is at least four times the R_(I) content of the region

(0°;0.4°).
 21. The retroreflective lens of claim 20, wherein the patternof retroreflection is such that the R_(I) content of the region

(0.4°;1.0°) is at least eight times the R_(I) content of the region

(0°;0.4°).
 22. A retroreflective lens comprising cube corners and havinga pattern of retroreflection such that the R_(I) content of the region

(0.3°;1.0°) is at least four times the R_(I) content of the region

(0°;0.3°) and is at least two times the R_(I) content of the region

(1.0°).
 23. A cube corner prism shaped in such manner that for axialincoming light, the retroreflected energy in each of two idealized spotsin a retroreflected spot pattern is no greater than one-half of theretroreflected energy in each of the other four idealized spots.
 24. Thecube corner prism of claim 23 for which the retroreflected energy ineach of two of the idealized spots is no greater than one-fourth of theretroreflected energy in each of the other four idealized spots.
 25. Aretroreflective lens comprising cube corners, said cube corners beingshaped in such manner that for axial incoming light the retroreflectedenergy in each of two idealized spots in a retroreflected spot patternis no greater than one-third of the retroreflected energy in each of theother four idealized spots.
 26. A retroreflective lens comprising aplurality of cube corner prismatic elements, wherein some elements aredisposed so as to be substantially symmetrical, except for theirdihedral angle errors, left to right, the substantially symmetricalelements producing a pattern of retroreflection comprising light moundssubstantially above and below center, but no light mound substantiallyleft or right of center. 27-28. (canceled)
 29. A retroreflective lenscomprising cube corners, most of said cube corners being oriented sothat there is an approximate left-right symmetry plane and most of thesecube corner elements having dihedral angle errors that are less thanabout 0.03 deg in absolute value for their one dihedral edge alignedwith said plane and also less than about 0.03 deg in absolute value forone of their other two dihedral edges while greater than about 0.10 degin absolute value for their third dihedral edge.
 30. A retroreflectivelens comprising cube corners, wherein a first plurality of cube cornerscomprise a first set of dihedral angle errors and first correspondingpattern of retroreflection and a second plurality of cube cornerscomprise a second set of dihedral angle errors, different from the firstset, and a second corresponding pattern of retroreflection, differentfrom the first pattern. 31-34. (canceled)
 35. A raised pavement markercomprising the lens of claim
 20. 36. A raised pavement marker comprisingthe lens of claim
 22. 37. A raised pavement marker comprising the prismof claim
 23. 38. A raised pavement marker comprising the lens of claim25.
 39. A raised pavement marker comprising the lens of claim
 26. 40. Araised pavement marker comprising the lens of claim
 29. 41. A raisedpavement marker comprising the lens of claim 30.